Schur Multiplication on B(lp,lq)

Quentin F. Stout
University of Michigan


Abstract: Let B(lp,lq) denote the bounded linear operators from lp to lq, for 1 ≤ p, q ≤ ∞. Termwise multiplication of two bounded linear operators in B(lp,lq) yields another bounded linear operator in B(lp,lq), called their Schur product (and sometimes misnamed their Hadamard product). The Schur product thus defines a commutative Banach algebra structure on B(lp,lq), even though when p ≠ q there was no natural product on this space, and when p = q there is a natural product but it is not commutative.

The maximal ideal spaces of these algebras are shown to be points in the Stone-Čech compactification of N x N, where N denotes the natural numbers. Various ideals and their hulls are examined, and several open questions are posed.

Keywords: Schur multiplication, Banach algebra, maximal ideal space, Stone-Cech compactification, operator theory, lp spaces, spectral synthesis

Complete paper. This paper appears in Journal of Operator Theory 5 (1981), pp. 231-243.

Here is some related work in operator theory.


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